Sensitivity measurement in TOC
We already know that in classical sensory psychophysics the concept of sensitivity is determined on the basis of the notion of the sensory threshold. The sensitivity value is understood as the reciprocal of the threshold value: the higher the threshold, the lower the sensitivity, and vice versa. Since all sensitivities in threshold psychophysics are reduced to measuring the threshold, there is no need to introduce any additional sensitivity indices. If the subject changes the decision criterion in the threshold estimation using the constant method, this means a simultaneous change in the threshold and, as a consequence, a change in sensitivity. Thus, the methodology of classical threshold psychophysics does not allow us to independently evaluate the processes associated with the influence on the criterion of decision-making of various cognitive and motivational factors, and the ability of the subject to detect a signal.
In the theory of signal detection, things are different. Here, sensitivity is understood as a value that reflects the signal-to-noise ratio in information processing channels. This value is considered as independent of the decision criterion, so that for the same criteria, the observer can show a different sensitivity, and conversely, the same sensitivity may correspond to different values of the criterion.
Formally, the sensitivity (denoted as d from English, detectability) in signal detection theory is defined as the difference between the expectations in the distribution of the sensory excitation of the signal in the noise and noise floor itself, expressed in units of the standard deviation for the distribution of noise effects. Mathematically, this definition can be expressed by the following formula:
So, if we get the value d ', in the experiment, say 1.50, this means that for the observer the distribution of the signal against the background of noise differs by one and a half unit of standard deviation, characterizing the distribution of noise.
The zero value d ' will mean that the observer in principle is not able to distinguish between noise and signal on its background. In other words, such a value d ' indicates that the affecting signal does not in any way change the background activity of the sensor systems that provide its detection. Note that, in spite of this, the subject can vary the number of positive and negative responses depending on the experimental conditions. However, changing the decision strategy in favor of omissions or false alarms will not result in a change in the effectiveness of the responses.
The situation is similar in the situation when the sensitivity value differs from the zero value. With the same value of noise and signal, d ' also remains unchanged when the number of hits and false alarms changes.
The work of the sensor system can be described graphically. Such a visual representation of the signal detection parameters was called the receiver performance (RXP).
The performance of the receiver is the ratio of the probabilities of hits and false alarms, which can be estimated in the experiment (Figure 7.2). The result of measuring the nature of the signal detection by the observer in this case is represented by a point on the graph
Fig. 7.2. The performance of the receiver reflects the ratio of the probabilities of hits and false alarms of the PFR .
If the subject is unable to isolate the signal from the noise, he, as we already know, relies on random guessing. It is clear that, regardless of how the subject establishes for himself the criterion for making a decision, the probability of hits and false alarms for him are equal in the general population, i.e. in theory. In this case, all points of the operating characteristic of the receiver are on the diagonal of the RHP, passing from the lower left corner to the right upper one. We call it an ascending diagonal.
The lower left corner of the RHP. where the rising diagonal originates, corresponds to the situation when the subject identifies all the samples submitted to him containing or without the stimulus sought exclusively as noise. In this case, he does not make false alarms, but the number of hits is zero. Such a decision-making strategy can be defined as extremely conservative. It guarantees no false alarms, but does not detect anything other than noise.
On the contrary, the upper right corner of the RHP, where the rising diagonal ends, corresponds to the situation when the subject uses an extremely careless, liberal, decision-making strategy, evaluating all the tests presented to him as signaling ones. This allows you to achieve the maximum of the correct hits, but as a consequence, is accompanied by the maximum number of false alarms, when all empty samples containing only noise are estimated as signaling.
Thus, we see that the position of the receiver performance point on the upward diagonal reflects solely the decision strategy of the observer, which sets the position of the decision criterion, and is in no way related to the characteristic of the ability of the sensor system to isolate the signal from noise. All points of the ascending diagonal correspond to zero sensitivity.
If the value d ' exceeds the zero value, it is obvious that the probability of hits will exceed the probability of false alarms (Figure 7.3). Thus, the result of the subject will be higher than the ascending diagonal of the RHP. Therefore, by the degree of remoteness from the result of the experimental result, the subject can be judged by how great his ability to isolate the signal from noise, i.e. how great is his sensitivity. However, this does not mean that the value of d ' can be judged solely from the absolute value of the remoteness of the PXP point from its diagonal.
To illustrate this point, consider Fig. 7.3. Here are the results of three measurements of the operating characteristics of the receiver. It can be seen that in all three experiments the position of the decision criterion was different. To see this, it suffices to compare the projections of three points onto the diagonal of the RXP. We see that in the first experiment the subject used the most conservative criterion. The number of hits, as well as the number of false alarms, is the least. In the third experiment, the subject uses the least cautious decision-making strategy. This leads to an increase in the number of hits, but at the same time, the number of false alarms also increases. In the second experiment, the decision strategy was the most balanced. However, the sensitivity in all sin experiments remained unchanged, despite the fact that the absolute distance between the points from the diagonal of the RHP varies. All three points lie on one curve, which is called the performance curve of the receiver.
Fig. 7.3. The PKP curve
Since all points of this curve correspond to the same sensitivity value, such a curve can be designated as a curve of equal sensitivity, or isosensitivity. There are an infinite number of such curves, and each of them corresponds to a certain sensitivity value. The more convex the shape of this curve, the larger d ' it corresponds (Figure 7.4).
Fig. 7.4. Isosensitivity curves
Thus, we see that on the basis of the data of the receiver operating characteristic and the isosensitivity curves, one can judge the position
the decision criteria during the detection of the signal, as well as the sensitivity value, reflecting how much, in principle, the observer is able to isolate the signal from the noise at their constant value. Thus, the operating characteristic of the receiver in the methodology of signal detection plays about the same role as the psychophysical function in classical threshold psychophysics. Nevertheless, just as in threshold psychophysics, in many cases it is important for the researcher to evaluate the values of the decision criterion and the sensitivity directly, ie, analytical, calculated, way.
It is clear that in practice the researcher has no idea about the nature of the noise distribution, even if he uses external sources of signal noise in the experiment. After all, in addition to external sources of noise, there are also internal sources associated with the operation of the sensory systems themselves. Therefore, it is impossible to estimate the sensitivity and the likelihood ratio corresponding to the criterion for making a decision using formulas (7.1) and (7.2). In addition, the position of the observer criterion does not necessarily correspond to the optimal value of the likelihood ratio.
The magnitude of the decision criterion can be established based on the probability of false alarms and hits. It can be given by the following relationships, where c - the size of the sought decision criterion:
But in order to solve these equations for c, it is necessary to again have an idea of the nature of the noise distribution. Suppose that it is described by the law of normal distribution. This assumption in most cases is very plausible and can easily be verified on the basis of available experimental data.
As is known, any normal distribution can be transformed on the basis of a linear transformation to a standard normal distribution, or z-distribution. Having carried out such a transformation for the noise distribution function, we have:
Thus, the value of the criterion can be obtained on the basis of z-transformations of the false alarm probability values:
If the noise distribution is described by a single normal distribution, then it is obvious that the quantity d ' should correspond to the mathematical expectation of the signal against the noise background, provided that this distribution is also normal and characterized the same variance:
Making a linear transformation of the signal distribution against the noise background by subtracting the value d ' from this distribution, we obtain the following relationship:
Hence, by performing z-transformations of the probability of hits, we have
Substituting the value with from equation (7.3) into the equation, we obtain a formula for calculating the sensitivity value d. Obviously, it can be obtained by the following formula:
(7.4)Knowing the position of the decision criterion, we can estimate the probabilities of the noise and signal values against the noise background. For this it is necessary to determine the ordinates of the noise and signal distribution functions against its background. Thus, we obtain a formula for calculating the likelihood ratio:
where O is the ordinate of the function of the standard normal distribution.
The likelihood ratio, more precisely, its logarithm (which in some cases may turn out to be more practical) can also be calculated directly from the results of the z-conversion of probabilities of hits and false alarms. To do this, you can use the following formula:
The advantage of calculating the logarithm (β) before calculation of the likelihood ratio itself is dictated primarily by convenience, since in this case the comparison is performed not with unity, but with zero.When choosing a balanced decision strategy, when the criterion is established in such a way that The probability that the observed sensory activity is caused by a signal at the noise foyer, and the probability that such activity is caused only by noise, is equal, the logarithm of p turns out to be 0. The negative value logarithm will testify in favor of a more liberal strategy of decision-making, while a positive value - in favor of the conservative.
In addition to the likelihood ratio β and its logarithm, other indexes are proposed in the theory of signal detection, which allow one to evaluate the position of the observer's criterion determining the prevalence of certain answers in the subject. Among them, the C index should be noted first. It can be defined as follows:
As you can see, this index is a derivative of lnβ. However, its calculation is somewhat simpler, since it does not need to be multiplied by d. That's why (and this is very important), its value does not depend on the value d '. Therefore, the calculation of this particular index is considered preferable. The value of C shows how many units of the standard deviation and in what direction from the intersection of the noise and signal distribution curves against the background is the criterion. If the criterion is located at the very intersection point of these distribution functions, the index C is zero.
Sometimes it is useful and important for the researcher to express the C index but the ratio of d. In this A value derived from C, which is commonly referred to as C ':
However, the value C ', is exactly the same as the likelihood ratio and its logarithm, depending on the sensitivity d'. This is the disadvantage of using this index.
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